Non-Perturbative Analysis of Physical and Mathematical Models

在对自然世界建模时，科学家们经常必须依赖近似，这会导致所研究系统的信息不完整。 基本粒子物理学、统计力学和引力就是这种情况。 因此，一个基本问题是如何从有限的数据中提取尽可能多的信息，并具有尽可能高的准确性和置信度。这个项目将利用数学的几个领域来分析数学和物理学中的这个问题。此外，研究人员的目标是进一步发展激光与物质相互作用的数学。该项目为研究生提供培训研究机会。该项目的一个中心主题是物质和辐射的相互作用。对于依赖于时间的激光诱导光电效应，研究人员将应用一种建设性的数学技术的光电离模型和实验中使用的双频模型，微调光电输出，并在跨学科的工作与实验学家，预测和优化的过程。该项目的第二个目的是确定扰动理论范围之外的物理和数学模型的行为，具有最高的准确性和置信度，当只有有限数量的扰动扩展项可以计算时。研究者的目标是开发出超过以前所达到的精确度的方法，并且在特定的数学条件下是最佳的。该方法也将被应用于数学问题，其中精确的解决方案是可用的，但其数值利用是计算的requirement.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

In modeling the natural world, scientists often must rely on approximations that result in incomplete information about a system under study. Such is the case in elementary particle physics, statistical mechanics, and gravitation. A fundamental question is then how to extract from limited data as much information as possible, with the possible highest level of accuracy and confidence. This project will draw on several areas of mathematics to analyze this issue in mathematics and physics. Additionally, the investigator aims to further develop the mathematics of laser-matter interaction. The project offers training research opportunities for graduate students. A central topic addressed in the project is the interaction of matter and radiation. For the time-dependent laser-induced photoelectric effect, the investigator will apply a constructive mathematical technique to the photoionization models and the two-frequency models used in experiments, to fine-tune the photoelectric output and, in interdisciplinary work with experimentalists, to predict and optimize the processes. A second aim of the project is to determine the behavior of physical and mathematical models outside the range of perturbation theory, with the highest possible level of accuracy and confidence, when only a limited number of terms of a perturbation expansion can be calculated. The investigator aims to develop methods that exceed the precision previously achieved and that are optimal under specific mathematical conditions. The methods will be applied also to mathematical problems in which exact solutions are available, but their numerical utilization is computationally demanding.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.